Introduction to Number Theory

Title Preface Contents I. Divisibility 1. Divisors 2. Remainders 3. Primes 4. The fundamental theorem 5. Least common multiple and greatest common divisor 6. Moduls, rings and fields 7. Euclid's algorithm 8. Relatively prime numbers. Euler’s phi-function 9. Arithmetical functions 10. Diophantine equations of the first degree 11. Lattice points and point lattices 12. Irrational numbers 13. Irrationality of the numbers e and pi Exercises II. On the distribution of primes 14. Some lemmata 15. General remarks. The sieve of Eratosthenes 16. The function pi(x) 17. Some elementary results on the distribution of primes 18. Other problems and results concerning primes III. Theory of congruences 19. Definitions and fundamental properties 20. Residue classes and residue systems 21. Fermat’s theorem and its generalization by Euler 22. Algebraic congruences and functional congruences 23. Linear congruences 24. Algebraic congruences to a prime modulus 25. Prime divisors of integral polynomials 26. Algebraic congruences to a composite modulus 27. Algebraic congruences to a prime-power modulus 28. Numerical examples of solution of algebraic congruences 29. Divisibility of integral polynomials with regard to a prime modulus 30. Wilson’s theorem and its generalization 31. Exponent of an integer modulo n 32. Moduli having primitive roots 33. The index calculus 34. Power residues. Binomial congruences 35. Polynomials representing integers 36. Thue’s remainder theorem and its generalization by Scholz Exercises IV. Theory of quadratic residues 37. The general quadratic congruence 38. Euler’s criterion and Legendre’s symbol 39. On the solvability of the congruences x^2 = -+2 (mod p) 40. Gauss’s lemma 41. The quadratic reciprocity law 42. Jacobi’s symbol and the generalization of the reciprocity law 43. The prime divisors of quadratic polynomials 44. Primes in special arithmetical progressions V. Arithmetical properties of the roots of unity 45. The roots of unity 46. The cyclotomic polynomial 47. Irreducibility of the cyclotomic polynomial 48. The prime divisors of the cyclotomic polynomial 49. A theorem of Bauer on the prime divisors of certain polynomials 50. On the primes of the form ny - 1 51. Some trigonometrical products 52. A polynomial identity of Gauss 53. The Gaussian sums Exercises VI. Diophantine equations of the second degree 54. The representation of integers as sums of integral squares 55. Bachet’s theorem 56. The Diophantine equation x^2 - D y^2 = 1 57. The Diophantine equation x^2 - D y^2 = -1 58. The Diophantine equation u^2 - D v^2 = C 59. Lattice points on conics 60. Rational points in the plane and on conics 61. The Diophantine equation a x^2 + b y^2 + c z^2 = 0 VII. Diophantine equations of higher degree 62. Some Diophantine equations of the fourth degree with three unknowns 63. The Diophantine equation 2 x^4 - y^4 = z^2 64. The quadratic fields K(sqrt(-1)), K(sqrt(-2)) and K(sqrt(-3)) 65. The Diophantine equation xi^3 + eta^3 + zeta^3 = 0 and analogous equations 66. Diophantine equations of the third degree with an infinity of solutions 67. The Diophantine equation x^7 + y^7 + z^7 = 0 68. Fermat’s last theorem 69. Rational points on plane algebraic curves. Mordell’s theorem 70. Lattice points on plane algebraic curves. Theorems of Thue and Siegel Exercises VIII. The prime number theorem 71. Lemmata on the order of magnitude of some finite sums 72. Lemmata on the Möbius function and some related functions 73. Further lemmata. Proof of Selberg’s formula 74. An elementary proof of the prime number theorem Exercises Tables Name index Subject index